Fast evaluation of elementary mathematical functions with correctly rounded last bit
ACM Transactions on Mathematical Software (TOMS)
SIAM Journal on Scientific Computing
Towards the Post-Ultimate libm
ARITH '05 Proceedings of the 17th IEEE Symposium on Computer Arithmetic
Elementary Functions: Algorithms and Implementation
Elementary Functions: Algorithms and Implementation
Emulation of a FMA and Correctly Rounded Sums: Proved Algorithms Using Rounding to Odd
IEEE Transactions on Computers
The pitfalls of verifying floating-point computations
ACM Transactions on Programming Languages and Systems (TOPLAS)
Handbook of Floating-Point Arithmetic
Handbook of Floating-Point Arithmetic
Midpoints and Exact Points of Some Algebraic Functions in Floating-Point Arithmetic
IEEE Transactions on Computers
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A very simple test, introduced by Ziv, allows one to determine if an approximation to the value f(x) of an elementary function at a given point x suffices to return the floating-point number nearest f(x). The same test may be used when implementing floating-point operations with input and output operands of different formats, using arithmetic operators tailored for manipulating operands of the same format. That test depends on a “magic constant” e. We show how to choose that constant e to make the test reliable and efficient. Various cases are considered, depending on the availability of an fma instruction, and on the range of f(x).